Electronic Journal of Probability

A Gaussian limit process for optimal FIND algorithms

Henning Sulzbach, Ralph Neininger, and Michael Drmota

Full-text: Open access

Abstract

We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to $c \cdot n^\alpha$ are chosen, where $0 < \alpha \leq \frac{1}{2}$, $c > 0$ and $n$ is the size of the data set to be split. We consider the complexity of FIND as a process in the rank to be selected and measured by the number of key comparisons required. After normalization we show weak convergence of the complexity to a centered Gaussian process as $n \to \infty$, which depends on $\alpha$. The proof relies on a contraction argument for probability distributions on càdlàg functions. We also identify the covariance function of the Gaussian limit process and discuss path and tail properties.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 3, 28 pp.

Dates
Accepted: 6 January 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065645

Digital Object Identifier
doi:10.1214/EJP.v19-2933

Mathematical Reviews number (MathSciNet)
MR3164756

Zentralblatt MATH identifier
1358.68085

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 68P10: Searching and sorting 60G15: Gaussian processes 60C05: Combinatorial probability 68Q25: Analysis of algorithms and problem complexity [See also 68W40]

Keywords
FIND algorithm Quickselect complexity key comparisons functional limit theorem contraction method Gaussian process

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Sulzbach, Henning; Neininger, Ralph; Drmota, Michael. A Gaussian limit process for optimal FIND algorithms. Electron. J. Probab. 19 (2014), paper no. 3, 28 pp. doi:10.1214/EJP.v19-2933. https://projecteuclid.org/euclid.ejp/1465065645


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